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MAT
MAT1503 (LINEAR ALGEBRA) EXAM QUESTIONS
WITH CORRECT VERIFIED SOLUTIONS 100%
GUARANTEED PASS (LATEST UPDATE)
Properties of the Transpose - ANS ✓If the sizes of the matrices are such that
stated operations can be performed, then
a) (Aᵀ)ᵀ = A
b) (A + B)ᵀ = Aᵀ + Bᵀ
c) (A - B)ᵀ = Aᵀ - Bᵀ
d) (kA)ᵀ = kAᵀ
e) (AB)ᵀ = BᵀAᵀ
The transpose of a product of any number of matrices is the product of the
transposes in the reverse order
Theorem 1.4.9 - ANS ✓If A is an invertible matrix, then Aᵀ is also invertible and
(Aᵀ)⁻¹ = (A⁻¹)ᵀ
row equivalent matrices - ANS ✓Either matrix can be obtained from the other
by a sequence of elementary row operations
Row Operations by Matrix Multiplication - ANS ✓If the elementary matrix E
results from performing a certain row operation on I and if A is an m×n matrix,
then the product EA is the matrix that results when this same row operation is
performed on A
Invertibility of Elementary Matrices - ANS ✓Every elementary matrix is
invertible, and the inverse is also an elementary matrix
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Equivalent Statements - ANS ✓If A is an n×n matrix, then the following
statements are all true or all false
a) A is invertible
b) Ax = 0 has only trivial solution
c) The reduced row echelon form of A is I
d) A is expressible as a product of elementary matrices
e) Ax = b is consistent for every n×1 matrix b
f) Ax = b has exactly one solution for every n×1 matrix b
g) det(A) ≠ 0
Inversion Algorithm - ANS ✓To find the inverse of an invertible matrix A, find a
sequence of elementary row operations that reduces A to the identity and then
perform that same sequence of operations on I to obtain A⁻¹
Theorem 1.6.1 - ANS ✓A system of linear equations has zero, one or infinitely
many solutions. There are no other possibilities
Theorem 1.6.2 - ANS ✓If A is an invertible n×n matrix, then for each n×1 matrix
b, the system of equations Ax = b has exactly one solution, namely, x = A⁻¹b
Theorem 1.6.3 - ANS ✓Let A be a square matrix.
a) If B is a square matrix satisfying BA = I, the B = A ⁻¹
b) If B is a square matrix satisfying AB = I, the B = A ⁻¹
Theorem 1.6.5 - ANS ✓Let A and B be square matrices of the same size. If AB is
invertible, then A and B must also be invertible
diagonal matrix - ANS ✓A square matrix in which all the entries off the main
diagonal are zero
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lower triangular matrix - ANS ✓A square matrix in which all the entries above
the main diagonal are zero
upper triangular matrix - ANS ✓A square matrix in which all the entries below
the main diagonal are zero
Theorem 1.7.1 - ANS ✓a) The transpose of a lower triangular matrix is upper
triangular, and the transpose of an upper triangular matrix is lower triangular
b) The product of lower triangular matrices, and the product of upper
triangular matrices is upper triangular
c) A triangular matrix is invertible if and only if its diagonal entries are all
nonzero
d) The inverse of an invertible lower triangular matrix is lower triangular,
and the inverse of an invertible upper triangular matrix is upper triangular
symmetric matrix - ANS ✓A square matrix A = Aᵀ
Theorem 1.7.2 - ANS ✓If A and B are symmetric matrices with the same size,
and if k is any scalar, then
a) Aᵀ is symmetric
b) A + B and A - B are symmetric
c) kA is symmetric
Theorem 1.7.3 - ANS ✓The product of two symmetric matrices is symmetric if
and only if the matrices commute
Theorem 1.7.4 - ANS ✓If A is an invertible symmetric matrix, then A⁻¹ is
symmetric
Theorem 1.7.5 - ANS ✓If A is a square invertible matrix, then AAᵀ and AᵀA are
also invertible
MAT1503
MAT
MAT1503 (LINEAR ALGEBRA) EXAM QUESTIONS
WITH CORRECT VERIFIED SOLUTIONS 100%
GUARANTEED PASS (LATEST UPDATE)
Properties of the Transpose - ANS ✓If the sizes of the matrices are such that
stated operations can be performed, then
a) (Aᵀ)ᵀ = A
b) (A + B)ᵀ = Aᵀ + Bᵀ
c) (A - B)ᵀ = Aᵀ - Bᵀ
d) (kA)ᵀ = kAᵀ
e) (AB)ᵀ = BᵀAᵀ
The transpose of a product of any number of matrices is the product of the
transposes in the reverse order
Theorem 1.4.9 - ANS ✓If A is an invertible matrix, then Aᵀ is also invertible and
(Aᵀ)⁻¹ = (A⁻¹)ᵀ
row equivalent matrices - ANS ✓Either matrix can be obtained from the other
by a sequence of elementary row operations
Row Operations by Matrix Multiplication - ANS ✓If the elementary matrix E
results from performing a certain row operation on I and if A is an m×n matrix,
then the product EA is the matrix that results when this same row operation is
performed on A
Invertibility of Elementary Matrices - ANS ✓Every elementary matrix is
invertible, and the inverse is also an elementary matrix
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Equivalent Statements - ANS ✓If A is an n×n matrix, then the following
statements are all true or all false
a) A is invertible
b) Ax = 0 has only trivial solution
c) The reduced row echelon form of A is I
d) A is expressible as a product of elementary matrices
e) Ax = b is consistent for every n×1 matrix b
f) Ax = b has exactly one solution for every n×1 matrix b
g) det(A) ≠ 0
Inversion Algorithm - ANS ✓To find the inverse of an invertible matrix A, find a
sequence of elementary row operations that reduces A to the identity and then
perform that same sequence of operations on I to obtain A⁻¹
Theorem 1.6.1 - ANS ✓A system of linear equations has zero, one or infinitely
many solutions. There are no other possibilities
Theorem 1.6.2 - ANS ✓If A is an invertible n×n matrix, then for each n×1 matrix
b, the system of equations Ax = b has exactly one solution, namely, x = A⁻¹b
Theorem 1.6.3 - ANS ✓Let A be a square matrix.
a) If B is a square matrix satisfying BA = I, the B = A ⁻¹
b) If B is a square matrix satisfying AB = I, the B = A ⁻¹
Theorem 1.6.5 - ANS ✓Let A and B be square matrices of the same size. If AB is
invertible, then A and B must also be invertible
diagonal matrix - ANS ✓A square matrix in which all the entries off the main
diagonal are zero
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lower triangular matrix - ANS ✓A square matrix in which all the entries above
the main diagonal are zero
upper triangular matrix - ANS ✓A square matrix in which all the entries below
the main diagonal are zero
Theorem 1.7.1 - ANS ✓a) The transpose of a lower triangular matrix is upper
triangular, and the transpose of an upper triangular matrix is lower triangular
b) The product of lower triangular matrices, and the product of upper
triangular matrices is upper triangular
c) A triangular matrix is invertible if and only if its diagonal entries are all
nonzero
d) The inverse of an invertible lower triangular matrix is lower triangular,
and the inverse of an invertible upper triangular matrix is upper triangular
symmetric matrix - ANS ✓A square matrix A = Aᵀ
Theorem 1.7.2 - ANS ✓If A and B are symmetric matrices with the same size,
and if k is any scalar, then
a) Aᵀ is symmetric
b) A + B and A - B are symmetric
c) kA is symmetric
Theorem 1.7.3 - ANS ✓The product of two symmetric matrices is symmetric if
and only if the matrices commute
Theorem 1.7.4 - ANS ✓If A is an invertible symmetric matrix, then A⁻¹ is
symmetric
Theorem 1.7.5 - ANS ✓If A is a square invertible matrix, then AAᵀ and AᵀA are
also invertible
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